A1: f is Function of X,X by FUNCT_2:121;

per cases ( X = {} or X <> {} ) ;

suppose A2: X = {} ; :: thesis: f . g is Element of X

then {} = dom f by A1, FUNCT_2:def 1;
then f . g = {} by FUNCT_1:def 4;
hence f . g is Element of X by A2, SUBSET_1:def 2; :: thesis: verum

end;

suppose X <> {} ; :: thesis: f . g is Element of X

hence f . g is Element of X by A1, FUNCT_2:7; :: thesis: verum

end;

end;